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Gibbs Measures with Multilinear Forms

arXiv (Cornell University)(2023)

Cited 0|Views13
Abstract
In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain necessary and sufficient conditions for replica-symmetry. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the ''local fields/magnetization'', the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest.
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要点】:本文研究了一类由广义U统计量给出的哈密顿量,以及一般基测度的多线性吉布斯测度,将渐进自由能表达为函数空间的优化问题,获得了复制度对称性的必要和充分条件,并得到了一大类感兴趣统计量的弱极限,包括局部场/磁化、哈密顿量、全局磁化等,并证明了在复制度对称性下,对比的普遍弱定律,以及温度参数意义上的尖锐相变点的存在性。

方法】:通过将渐进自由能表达为函数空间的优化问题,并利用该优化问题获得了复制度对称性的必要和充分条件。

实验】:使用本文提出的方法,对局部场/磁化、哈密顿量、全局磁化等多种统计量进行了弱极限分析,并证明了在温度参数意义上的尖锐相变点的存在性。此外,还证明了局部和全局磁化的指数集中界限。